I have a Stored Procedure which is triggered to import/process a bunch of data from an import table. Within the SP is a UPDATE statement which it appears to not get triggered, however all the other statements within the stored procedure run fine.

At the moment if I want this UPDATE statement to run, I have to open the stored procedure within the SQL Server Management tool highlight the UPDATE statement and execute it. All checks show that the UPDATE statement has done what I expected it to do.

You don't have to; you can debug it yourself as follows: Insert a few print statements at points from the beginning of the stored proc to just before the update statement is called. Then run the stored proc from an SSMS window and see which of those get printed. I suspect the one just before the update statement won't get printed.